The dichotomy between chemical concepts and numbers after almost 100 years of quantum chemistry: conceptual density functional theory as a case study

The dichotomy whether in Quantum Chemistry insight and numbers are to be placed on equal footing is situated in a historical perspective starting from Coulson’s famous quote “Give us insight, not numbers” to Neese’s recent adaptation “Give us insights and numbers”. In parallel, the problem of the chemical interpretation of complex computational results in terms of classical chemical concepts is addressed starting from Mulliken’s quote that “the more accurate the calculations became the more concepts tend to vanish in the air”. Conceptual Density Functional Theory is one of the techniques which avoids the latter issue by its density based approach which can be applied to computational results of any level of sophistication. The crucial role of response functions of the energy E with respect to perturbations in the number of electrons N and/or external potential v ( r ) is thereby highlighted. However a new confrontation “insight versus numbers” appears. When gradually refining the evaluation of these descriptors, in order to pass from a qualitative to a quantitative level, it turns out from our previous studies that either minor influences show up, or that also fundamental issues may arise hidden in the definition and the physical background of the response function. The derivative discontinuity of the E vs. N curve hereby plays a fundamental role. This issue is documented scrutinizing a series of recent studies on the analytical evaluation of the three second order response functions: the linear response function, the Fukui function and the chemical hardness. They are shown to behave in a fundamentally different way under refinement. For the linear response function, the pure second order v functional derivative of the energy, no fundamental problems arise: when passing to a full analytical evaluation an increasing level of complexity of the equations is observed leading however to a smooth convergence. In the Fukui function case, involving a mixed N and v energy derivative, the issue with the E = E ( N ) curve and N derivative can be circumvented by directly deriving the electronic chemical potential with respect to v using a Maxwell type relation. Finally for the chemical hardness involving the pure second order N derivative, a fundamental problem arises due to the derivative discontinuity when refining the venerable Parr-Pearson parabolic E = E ( N ) curve. It forces us to stick to its result identifying the hardness as the (band) gap. On the other hand the analytical expression yields a “condition” that Density Functional Approximations should obey. It is shown how its implementation leads to a straightforward estimate of their delocalization error, on the road for further improvement of DFAs. The inclusion of temperature may be a way out for further refining the chemical hardness and all other response functions involving second or higher order N derivatives, the simplest case being the dual descriptor. Overall this evolution reflects the basic characteristics Löwdin’s accuracy vs. refinement graph.